Sampling Distribution - Purdue University

Lecture Notes of Stat 355, Sampling Distributions

Sampling Distribution

Tonglin Zhang

Tonglin Zhang, Department of Statistics, Purdue University

Sampling Distributions

Well Known Distributions

Well Known Distributions

We want to use computers to understand the following well known distributions.

Discrete distributions. In practice, it can only be integers and mostly nonnegative.

Binomial. Poisson.

Continuous distributions. In practice, it can only be values within an interval, including (-, ). Uniform. Normal. Exponential. Chi-square, t, and F . Gamma and Beta.

Tonglin Zhang, Department of Statistics, Purdue University

Sampling Distributions

Well Known Distributions

Let X be the random variables from the distribution. We need

How to generate X with n independent replications, called samples.

The computation of the mean and sample variance based on

the sample. For instance, let x1, ? ? ? , xn be the sample. Then,

the mean is

1 n

x? = n

xi

i =1

and the sample variance is

s2

=

n

1 -

1

n (xi

-

x?)2,

i =1

where the standard error is s.

Tonglin Zhang, Department of Statistics, Purdue University

Sampling Distributions

Well Known Distributions

PMF (probability mass function) for discrete or PDF (probability density function) for continuous.

CDF:

F (t) = P(X t)

for any fixed t. Quantile: the inverse of CDF. Find the value t such that

P(X t) = q

for any fixed q (0, 1), often expressed as t = F -1(q).

I am going to talk these based on specific distributions.

Tonglin Zhang, Department of Statistics, Purdue University

Sampling Distributions

Well Known Distributions

Uniform

If the probability is only proportional to the length of the interval, then it is the uniform distribution.

Suppose that values can only be within the interval [1, 2]. Then, for any (a, b) [1, 2], there is

P{X (a, b)} = b - a , 2 - 1

which is denoted by

X Uniform[1, 2].

The standard case is 1 = 0 and 2 = 1. Please read my R code to understand (1) samples (2) CDF

(3) quantile.

Tonglin Zhang, Department of Statistics, Purdue University

Sampling Distributions

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